methods for evaluating the temperature structure-function ...weather.ou.edu/~fedorovi/pdf/ef papers...

Boundary-Layer Meteorol (2015) 155:189–208DOI 10.1007/s10546-014-0001-9

ARTICLE

Methods for Evaluating the TemperatureStructure-Function Parameter Using UnmannedAerial Systems and Large-Eddy Simulation

Charlotte E. Wainwright · Timothy A. Bonin ·Phillip B. Chilson · Jeremy A. Gibbs ·Evgeni Fedorovich · Robert D. Palmer

Received: 4 August 2014 / Accepted: 29 December 2014 / Published online: 9 January 2015© Springer Science+Business Media Dordrecht 2015

Abstract Small-scale turbulent fluctuations of temperature are known to affect the propaga-tion of both electromagnetic and acoustic waves. Within the inertial-subrange scale, wherethe turbulence is locally homogeneous and isotropic, these temperature perturbations canbe described, in a statistical sense, using the structure-function parameter for temperature,C2

T . Here we investigate different methods of evaluating C2T , using data from a numerical

large-eddy simulation together with atmospheric observations collected by an unmannedaerial system and a sodar. An example case using data from a late afternoon unmanned aerialsystem flight on April 24 2013 and corresponding large-eddy simulation data is presentedand discussed.

Keywords Large-eddy simulation · Sodar · Structure-function parameter ·Unmanned aerial system

1 Introduction

Temperature fluctuations associated with atmospheric turbulence can strongly affect the prop-agation of electromagnetic and acoustic waves in the atmosphere (Tatarskii 1961). Under-standing the impact of these fluctuations on wave propagation and scattering is importantfor the correct interpretation of returns from remote-sensing instruments such as radars andsodars. An improved understanding of the spatial variability of temperature allows for betterinterpretation of sodar and radar returns, and thus maximizes the extractable informationfrom these returns.

C. E. Wainwright (B) · P. B. Chilson · R. D. PalmerAdvanced Radar Research Center and School of Meteorology,University of Oklahoma, Norman, OK, USAe-mail: [email protected]

T. A. Bonin · J. A. Gibbs · E. FedorovichSchool of Meteorology, University of Oklahoma, Norman, OK, USA

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190 C. E. Wainwright et al.

A useful parameter for quantifying the spatial variability of temperature in the atmosphereis the structure function for temperature, (δT )2, which is calculated as

(δT )2(r, t) = [T (x, t) − T (x + r, t)]2, (1)

where the overbar represents ensemble averaging, T is temperature, x is a position vector, ris a separation vector, and t is time. If the separation distance r = |r| is within the inertialsubrange of turbulence scales, where the temperature fluctuations are locally isotropic, thenthe temperature structure function can be represented as

(δT )2 = C2T r2/3, (2)

where C2T is the structure-function (or structure) parameter for temperature (Tatarskii 1971).

Knowledge of the temperature structure-function parameter is important not only for betterinterpretation of instrument returns, but also because the refractive index structure-functionparameter, C2

n , is related to C2T in monostatic configurations via

C2n ∝ C1C2

T + C2CT q + C3C2q , (3)

where CT q is the temperature-humidity structure-function parameter and C2q is the structure-

function parameter for humidity, and C1, C2, and C3 are constants whose value vary foracoustic and electromagnetic scattering (following Tatarskii 1961, 1971; Wyngaard et al.1971).

Knowledge of the vertical profiles of C2n is important for a wide range of topics, including

optics and ground-to-satellite communications (e.g., Tunick 2005). Fluctuations in opticalwave-propagation parameters are of importance for astroclimatic studies, as these fluctuationsdepend upon C2

n . Hence C2n fields are used to determine optimal sites for astronomical

observatories (e.g., Gur’yanov et al. 1992; Travouillon et al. 2003; Petenko et al. 2014a).Experimental methods for evaluating C2

T have typically involved ground-based temper-ature measurements (e.g., Kunkel et al. 1981), thermometers mounted on tethered balloonsor radiosondes (e.g., Readings and Butler 1972; Balsley 2008), or airborne measurements(e.g., Thomson et al. 1978). These in situ methods require either two sets of simultaneouslyrecorded temperature measurements separated by a known distance, or high-temporal reso-lution measurements of temperature, wind speed and direction, plus invoking Taylor’s frozenfield turbulence hypothesis (Taylor 1938).

Remote sensing instruments can also be used to derive C2T . Acoustic remote sensing has

previously been used to evaluate C2T through use of the sodar returned power (Neff 1975).

Many previous studies have compared C2T derived from sodars to those directly evaluated by

methods mentioned in the previous paragraph, finding fair agreement (e.g., Asimakopouloset al. 1976; Weill et al. 1980; Gur’yanov et al. 1987; Kumar et al. 2011). A comprehensiveanalysis of the statistical characteristics of variations in sodar-derived C2

T under convectiveconditions can be found in Petenko et al. (2014b).

Scintillometry can also be used to derive C2T values (e.g., Frehlich 1992), since the refrac-

tion of light in the atmosphere is primarily controlled by temperature fluctuations. Scintil-lometers provide path-averaged C2

T values that typically represent spatial scales of temper-ature fluctuations from tens of metres to a few kilometres. Previous studies have shown thatscintillometer-derived C2

T values compare well with those estimated from sonic-anemometertemperature and wind measurements (e.g., Wood et al. 2013).

More recently, unmanned aerial systems (UASs) have become a widely-employed tech-nique for meteorological studies (e.g., Holland et al. 2001; Shuqing et al. 2004; Spiess et al.

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Evaluating C2T from LES and UAS 191

2007; van den Kroonenberg et al. 2008). Since UAS platforms can provide high-resolutiontemperature, wind, and location data, they are well placed to examine C2

T . This capabilityhas recently been explored by van den Kroonenberg et al. (2012), who used a UAS to studyC2

T over a heterogeneous land surface on two summer days. C2T values were derived from

temperature data recorded during straight-line and square-path flights, and then comparedto C2

T derived from sonic-anemometer measurements. Good agreement was found in thediurnal variation in C2

T measured by the sonic thermoanemometers and UAS.Aside from using measurements from in situ or remote sensing instruments, C2

T has alsobeen examined through numerical simulation. The advancement of the large-eddy simulation(LES) technique has provided a tool to examine structure-function parameters of atmosphericflow in four dimensions both from LES data (Peltier and Wyngaard 1995; Cheinet and Cumin2011; Wilson and Fedorovich 2012; Maronga et al. 2013), and through the development ofthe LES-based numerical radar simulators (Muschinski et al. 1999; Scipión et al. 2008). Oneadvantage of using LES data for this purpose is that both the spatial and temporal variabilityof C2

T can be examined across a range of scales and also in relation to the separation-distancedependence. In addition, the use of LES allows for the testing of different methods to deriveC2

T from the same data, something that cannot be easily achieved in the field.The present work examines how UAS can be employed to evaluate the structure-function

parameter for temperature, and proposes a UAS simulator capable of ingesting LES data. Thisallows for a comparison of theoretical UAS C2

T retrievals with those from the LES generatedusing different retrieval algorithms. Our study also compares C2

T measurements derivedfrom a UAS and a sodar operated at the University of Oklahoma’s Kessler Atmospheric andEcological Field Station (KAEFS). The KAEFS facility is a heavily instrumented field sitelocated in rural Oklahoma, encompassing a mixture of rolling grassland and heterogeneousvegetation. A LES experiment was conducted with the domain centered upon the field site andthe time period encompassing that during which the UAS and sodar were operational. TheLES data, in the form of highly temporally- and spatially-resolved flow fields, is used to testthe UAS simulator, and also to directly evaluate C2

T . This allows for a sensible comparisonof several methods of deriving C2

T and an examination of the effects of spatial and temporalaveraging on the derived values.

The rest of the paper is organized as follows: in Sect. 2, the UAS platform used for theexperiment is described. Details on the UAS simulator are presented in Sect. 3. Section 4explains the methods used to extract C2

T from the LES data, from the UAS simulator and realUAS data, and from the sodar. A description of the experiment conducted using LES data inconjunction with UAS, and results from the experiment are presented in Sect. 5. Conclusionsare provided in Sect. 6.

2 Unmanned Aerial System

The University of Oklahoma operates an unmanned aerial system known as the Small Mul-tifunction Autonomous Research and Teaching Sonde (SMARTSonde, Bonin et al. 2012,2013). The SMARTSonde is primarily used to investigate the structure of the atmosphericboundary layer. The particular airframe employed in this study was a Funjet Ultra, which hasa wingspan of 0.8 m and a weight of approximately 1 kg, an endurance of 30 min, and a cruis-ing speed of 15 m s−1. The SMARTSonde is equipped with sensors to measure atmospherictemperature, pressure, and relative humidity; the response time for the pressure sensor is 0.5 s,accurate to ±1.5 Pa. The response time and measurement accuracy for the relative humid-ity sensor are 8 s and ±1.8 %, respectively, while the accuracy of the temperature sensor

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measurement is ±0.3 K. The time constant of the sensor was calculated through calibration,resulting in a time constant of 3 s for an airspeed of 15 m s−1.

The SMARTSonde employs the open source Paparazzi autopilot system for autonomousflight, although a pilot with a radio controller manually operates the aircraft during take-offand landing. A more complete description of the UAS package can be found in Bonin etal. (2013). As an update, an inertial measurement unit has recently been integrated into theautopilot for determination of the aircraft altitude.

3 UAS Simulator

A UAS simulator was developed in order to examine the relative merits of different methodsof retrieving C2

T from UAS measurements. The simulator ingests three-dimensional fields oftemperature from large-eddy simulation (LES) output, which are sampled in a manner thatmimics the flight plan of the UAS in the atmosphere.

Many aspects of the UAS simulator are defined by the user, in the way that the simulatorcan be used to emulate desired flight plans. The SMARTSonde utilizes two main flightpatterns (plans) for boundary-layer research: the first is a steady helical ascent, which isused alongside specially designed retrieval algorithms to derive the wind speed and direction(Bonin et al. 2013). The second flight pattern is applied to examine the spatial structure andstatistical properties of thermodynamic quantities such as temperature and humidity. Thisflight plan entails the SMARTSonde flying in a number of consecutive circles at each heightof interest before ascending to the next height (as illustrated in Fig. 1). The UAS simulatorcan mimic both of these flight plans, but it is the second plan (Fig. 1) that is used in thecurrent study.

User-defined flight-plan characteristics in the UAS simulator include the circle radius,the desired airspeed, the height interval at which to sample, and the amount of time for thesimulated UAS to remain at each height before ascending. These flight characteristics arethen translated into Cartesian coordinates within the LES domain at times corresponding toeach measurement (based upon the desired flight speed). The LES temperature and winddata are spatially and temporally interpolated to provide temperature and horizontal windcomponent (u in the x direction and v in the y direction) measurements at each samplingpoint. Fields of u and v are required to enable advection correction, which is discussed inSect. 4.2. The structure-function parameter for temperature is evaluated using these virtualtemperature and wind measurements, using the method described in Sect. 4.2.

Fig. 1 The flight path used bythe UAS for measuringtemperature, which can be usedto calculate C2

T

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4 Methods for Evaluating C2T

Four methods of evaluating C2T are utilized herein: direct evaluation from LES output; evalua-

tion using a simulated UAS flight within the LES domain; derivation from UAS data recordedduring a field experiment; and derivation from sodar return power. Thorough descriptions ofeach of these methods are provided in the following subsections.

4.1 Evaluating C2T from LES Data

In evaluating C2T from LES data, only the four-dimensional (three spatial coordinates and

time) temperature field is required, along with knowledge of the grid spacing and timeresolution of the generated field. The method calculates a running sum of squared temperaturedifference values for a given separation distance in a specific direction and within a horizontalcross-section of the domain located at a particular height. Once squared differences from theentire horizontal plane have been summed, an average is calculated. This procedure is repeatedfor each height and time before proceeding to the next separation distance. This method isexplained in detail in Wilson and Fedorovich (2012) for the refractive index n.

The squared temperature differences (δT )2 are calculated in four separate directions: x(zonal differences), y (meridional differences), the positive diagonal (henceforth referred toas xy), and the negative diagonal (henceforth yx). These calculations are performed as

(δT )2|x (x, xi , y, z, t) = [T (x, y, z, t) − T (x − xi , y, z, t)]2 , (4)

(δT )2|y(x, y, yi , z, t) = [T (x, y, z, t) − T (x, y − yi , z, t)]2 , (5)

(δT )2|xy(x, xi , y, yi , z, t) = [T (x, y, z, t) − T (x − xi , y − yi , z, t)]2 , (6)

and(δT )2|yx (x, xi , y, yi , z, t) = [T (x, y, z, t) − T (x − xi , y + yi , z, t)]2 . (7)

In Eqs. 3–6, xi is the separation distance in the x-direction and yi is the separation distance inthe y-direction. Note that in Eqs. 5 and 6, xi = yi , such that the squared temperature differencesare calculated on strict diagonals only. All the separation distances used to calculate (δT )2|xand (δT )2|y are multiples of the LES horizontal grid spacing, � = �x = �y, so nointerpolation between numerical grid points in the horizontal plane is needed. The separationdistances used to calculate (δT )2|xy and (δT )2|yx are multiples of the minimal diagonaldistance (�xy = √

�x2 + �y2).The method outlined above is illustrated in Fig. 2. The upper plane illustrates the calcula-

tion of (δT )2|xy (Eq. 5) for separation distances of xi = yi = �, 2� and 3�, and the lowerplane shows the calculation of (δT )2|y for separation distances of yi = �, 2� and 3�.This method is applied in Sect. 5 to the LES data described in Sect. 5.1, and the results aredescribed in Sect. 5.2. The LES model used to test the method has a grid spacing of 5 m. The(δT )2|x values are calculated for each separation distance xi = i� where i = 1, 2, . . . , 50,and for each height and time of interest. The maximum value of i was set to 50 to give a max-imum separation distance r of 250 m, which is a typical diameter of the UAS flight circles.The calculated (δT )2|x values are then spatially averaged for each separation distance i�.This method is repeated for Eq. 4 using separation distances yi = i� where i = 1, 2, …, 50 asbefore, and for Eqs. 5 and 6 using xi = yi = i� where i = 1, 2,…, 50 (equating to straight-line separation distances of i�xy = √

2 × 12,√

2 × 22, . . . ,√

2 × 502). The correspondingstructure-function counterparts to the squared differences in Eqs. 3–6 are calculated as

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Fig. 2 Illustration of themethods of evaluating C2

Tdirectly from the LES. The lowerz level illustrates the (δT )2|ymethod and the upper z levelshows the (δT )2|xy method

y

z

x

ii = 2 i

i = 2

i = 3

(δT )2|x (z, i�) = 〈(δT )2|x (x, xi , y, z, t)〉x,y,t , (8)

(δT )2|y(z, i�) = 〈(δT )2|y(x, y, yi , z, t)〉x,y,t , (9)

(δT )2|xy(z, i�xy) = 〈(δT )2|xy(x, xi , y, yi , z, t)〉x,y,t , (10)

and(δT )2|yx (z, i�xy) = 〈(δT )2|yx (x, xi , y, yi , z, t)〉x,y,t , (11)

where the angled brackets denote the averaging within each horizontal plane and in time.Conceptually, the notion of C2

T makes sense only when the turbulence is assumed to beisotropic and homogeneous within the inertial subrange of spatial scales. Use of LES allowsfor examination of the validity of the isotropy assumption by comparing the (δT )2 valuesfor different directions. If the turbulence is close to the isotropic state within certain scaleranges, there should be only minor differences between the values of (δT )2 values calculatedusing Eqs. 7–10 for the corresponding ranges of the separation distances.

4.2 Evaluating C2T from the UAS Simulator

The UAS simulator provides T , u, and v values corresponding to the simulated location ofthe UAS within the LES domain for the duration of the simulated flight. For each individualmeasurement height, the temperature data are combined to include all possible data pairs(providing n(n − 1)/2 unique pairs of temperature data, where n is the total number oftemperature measurements recorded at that height). A squared temperature difference, (δT )2,can then be calculated from each temperature pair.

As the simulated UAS coordinates are known at every time, the separation distancebetween the two data points, r , is then calculated for each temperature pair. The calcula-tion of the temperature difference pairs and separation distance is illustrated in Fig. 3. Theseparation distances are modified to account for the advection of turbulence past the UASby the wind, as detailed in Bonin et al. (2015). With advection correction, the separationdistance is calculated as

r =√

(xi − u�t)2 + (yi − v�t)2, (12)

where xi and yi are the separation distances in the zonal and meridional directions, u and v

are the instantaneous zonal and meridional wind speeds in m s−1, and �t is the time betweenconsecutive measurements. Advection correction is not performed when C2

T is evaluateddirectly from the LES as described in Sect. 4.1, since in this case all the temperature valuesat a given height correspond to the same moment of time. However, the real and simulated

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Fig. 3 Illustration of themethods of evaluating C2

T fromthe UAS flight and simulatedflight. Only eight data pointsaround the circle are used forclarity. In this example, C2

T iscalculated at four separationdistances, 153 m (solid blackline), 283 m (red line), 370 m(blue line), and 400 m (dashedblack line). The radius of thecircular flight path is 200 m

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The temperature pairs are separated into bins based on r2/3, and the structure function,(δT )2, is calculated by averaging all the (δT )2 values within each bin. The structure-functionparameter for temperature, C2

T , is then evaluated for each bin using Eq. 2.

4.3 Evaluating C2T from UAS Data

The method used for evaluating (δT 2), and subsequently C2T , from data recorded by the UAS

is very similar to that described in the previous sub-section for the UAS simulator. The spatialcoordinates of the UAS during the flight are recorded by a global positioning system (GPS)sensor at the same temporal rate that the temperature is recorded. The wind components uand v can be estimated using the heading direction, throttle, and instantaneous airspeed datarecorded by the UAS. The method used to estimate C2

T from the UAS data is outlined indetail in Bonin et al. (2015).

The recorded temperature data are divided into height batches based upon the heightrecorded by the GPS. This is done manually to ensure that time periods during take-off, finaldescent, landing, and during which the plane is ascending from one height to the next, areexcluded. The recorded locations, temperatures, and derived wind components are then usedto calculate (δT 2) and C2

T values as outlined in Sect. 4.2.

4.4 Evaluating C2T from Sodar Data

The method of retrieving C2T from sodar data makes use of the returned sodar power (as

described in, e.g., Weill et al. 1980),

PR = PT G Aeσscτ

2

exp(−2αh)

h2 , (13)

where PT is transmitted power, G is the gain of the antenna, Ae is the antenna effectivearea in m2, σs is the acoustic backscattering coefficient at 180◦ in m−1, c is the speed of

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sound in m s−1, τ is the acoustic pulse duration in s, h is the range of the signal in m, and α

represents acoustic attenuation due to atmospheric absorption. It should be noted that manycommercially available sodars scale the returned power and then present the data in units of dBwith an arbitrary zero. This is done to minimize the storage space required for the data, sinceunscaled values of PR are typically on the order of 10−16 W. The effective backscatteringcross-section per unit of scattering volume per unit of solid angle, σs , is related to C2

T as

σs = 4 × 10−3λ−1/3 C2T

T 2 , (14)

where λ is the sodar wavelength in m, and T is in K (following Little 1969). The sodarprovides, as output data, values of PR and h, whilst PT , G, Ae, and τ are functions of thesodar system used. The value of λ is indirectly set by the sodar user, and is a function of theacoustic transmit frequency, f , and the speed of sound, c, calculated as λ = c/ f .

The acoustic attenuation α is calculated according to ISO9613-1 (1993). The relation for α

is not reproduced here for brevity. The full calculation of α requires input of vertical profilesof temperature T and specific humidity q encompassing the range over which α is to becalculated, and the acoustic transmit frequency f . Since the sodar does not measure relativehumidity, additional instrumentation is needed to provide these data. In our case, the UAS canprovide the necessary temperature and humidity data. In order to evaluateα, the T and q valuesobtained from the UAS are interpolated to the sodar measurement heights. This naturallyrestricts the calculation of α, and thus C2

T , to heights within the range encompassed by theUAS. For the considered case, the average temperature and relative humidity values were13 ◦C and 35 %, which results in attenuation due to atmospheric absorption of approximately0.016 dB m−1 for the sodar frequency of 1,900 Hz.

The speed of sound, c, is calculated based upon the UAS temperature and humiditymeasurements according to

c = √γa Rd Tav, (15)

where γa is the specific heat ratio of air, Rd is the gas constant for dry air (287 J K−1 kg−1),Tav is the acoustic virtual temperature in K, and c is in m s−1 (Kaimal and Businger 1963). Tav

can be calculated from the temperature T and specific humidity q as Tav = T (1 + 0.513q).The value of the backscattering coefficient σs can then be estimated using (12), while thesodar-estimated C2

T can then be evaluated through Eq. 13.

5 Example Case

5.1 Experiment Description

On April 24 2013, a field experiment was conducted at the KAEFS site. The UAS performedseveral flights using a flight plan of repeating circles at prescribed heights, as illustrated inFig. 1. A Metek PCS.2000 sodar located approximately 150 m from the centre of the UASflight path was in operation for the duration of the UAS flights. Three flights were performed,and here we use data recorded during the second flight, which took place from 1824 to 1855local time (LT; 2324–2355 UTC). The experiment time window covers the start of the earlyevening transition regime, in which the turbulence in the boundary layer begins to decay.Further details about the SMARTSonde, the flights performed during this experiment, andthe calculation of C2

T from the UAS data can be found in Bonin et al. (2015).

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A large-eddy simulation was performed for the time period 1700–1900 LT on April 24,corresponding to the time of the UAS flights. The University of Oklahoma LES (OU-LES,Fedorovich et al. 2004a) was used in the experiment. The OU-LES code has been testedextensively for convective boundary layer (CBL) conditions, and it has been proven to real-istically reproduce mean flow and turbulence structure under clear CBL conditions such asobserved during the experiment (Fedorovich et al. 2001, 2004b; Gibbs et al. 2011).

The simulation was performed using a variable timestep (based upon numerical stability),with an average timestep during the second hour of simulation being 0.4 s. The effectivetemporal resolution of the OU-LES is 5–7�t , which gives an effective time constant of theLES of about 2.5 s. The domain size was X × Y × Z = 640 m × 640 m × 3 km, with auniform grid spacing of �x = �y = �z = 5 m. The lowest numerical grid level was locatedat 2.5 m above the ground. The employed subgrid turbulence closure scheme is based uponDeardorff (1980). The LES was nudged with output from a Weather Research and Forecast-ing (WRF, Skamarock et al. 2008) model run with a 4-km horizontal grid spacing, centeredover the KAEFS site. The WRF model was initialized using data from the North AmericanRegional Reanalysis (NARR, Mesinger et al. 2006) fields. Horizontal mean profiles of thezonal and meridional winds u and v, the specific humidity q , and the potential temperature θ

were taken from the WRF output every minute. These profiles were temporally interpolatedand used to nudge the corresponding prognostic LES fields at every timestep, as explained inGibbs et al. (2011).

The mean profiles of potential temperature from the LES and UAS and the horizontal windcomponents from the LES, UAS, and sodar are illustrated in Fig. 4. The LES is seen to produceslightly warmer atmospheric conditions than sensed by the UAS, although the potentialtemperature variation with height is very similar in the LES data and UAS measurements. Thewind profiles are also seen to be broadly similar, with slightly higher wind speeds measuredby the UAS than predicted by the LES at low heights. The mean wind profiles from thesodar are also presented, but the large variation between averaging periods encompassingthe flight period indicate that the sodar wind data may be unreliable. For further discussionof the atmospheric conditions during the field experiment see Bonin et al. (2015), who also

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compare wind speed data from a lidar located at the experimental site, which matches thatfrom the LES more closely than the sodar data.

The UAS simulator was populated with turbulent flow fields from the described LESfor the time period 1825–1850 LT, encompassing the duration of the ascending portion ofthe UAS flight. The simulated flight plan included circles of radius 200 m, at 50-m heightincrements from 50 to 500 m above the ground (as illustrated in Fig. 1). Two successivecircles were simulated at each separate height, with 48 data points collected equidistantlyaround each circle. The relative ground speed of the simulated UAS for this experiment isapproximately 18 m s−1. Unlike the real UAS, there was no time delay since the simulatedUAS moved from one height up to the next. The ground-relative speed of the simulatedUAS also remained unaffected by external factors such as wind speed. Temperature timeseries from the simulated UAS are illustrated in Fig. 5, where the temperature decreasewith height is clearly seen. Figure 5 also shows the relative temperature variation at eachheight, where the left panel of the figure indicates larger temperature variations during thefirst few minutes (i.e., at the lowest measurement height) than during the later portion of theflight.

The method outlined in Sect. 4.2 was used to calculate C2T from the simulated UAS

temperature data. A similar method (outlined in Sect. 4.3) was also used on the recordedtemperature data from the actual UAS flight. The structure-function parameter values derivedfrom the UAS simulator were then compared with their counterparts derived directly fromthe LES temperature field, while the C2

T values derived from the UAS flight were comparedwith those derived from the sodar data.

5.2 Results

It is known that the concept of C2T applies only to locally isotropic turbulence within the

inertial subrange of spatial scales. To examine the isotropy of the turbulence over horizontalplanes, we compare the four sets of (δT 2) values: (δT 2)|x , (δT 2)|y , (δT 2)|xy , and (δT 2)|yx .

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100 101 102 103 10410−4

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)2[K

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(δT )2|yx

r 2/3

Fig. 6 Variation of (δT 2) with separation distance at a height of 200 m from the four direction evaluateddirectly from LES output. The black lines mark the bounds between which the turbulence from the LES iswithin the inertial subrange

Figure 6 shows the variation of (δT 2) along different directions with separation distance,for a height above the ground of 200 m. Examining Fig. 6, it can be seen that there is goodagreement between (δT 2)|x , (δT 2)|xy , and (δT 2)|yx , whilst (δT 2)|y shows slightly smallerC2

T values than the other three directions, most notably at larger separation distances. An

investigation into the discrepancy between (δT 2)|y and (δT 2) evaluated in the other threedirections suggested that wind shear is likely responsible for this effect. The simulationcontained non-zero wind shear, with a mean value of v across the domain at a height of 200 mbeing −2.7 m s−1 and a corresponding value of u of −0.5 m s−1. Wind shear acts to elongateturbulent eddies in the dominant wind direction, which reduces temperature differences inthis direction (that is, reducing (δT 2)|y in this case study). The role of a sheared environmentin introducing turbulence anisotropy has been demonstrated for velocity spectra in Gibbs andFedorovich (2014).

To further examine whether the simulated turbulence is within the inertial subrange ofscales, the inertial subrange r2/3 separation distance dependence, Eq. 2, is evaluated. Theblack dashed line in Fig. 6 represents the r2/3 dependence. Comparing this line to the four(δT 2) dependencies one may notice that for separation distances between approximately 20and 160 m, marked by the vertical lines in Fig. 6, (δT 2) does indeed follow the r2/3 lawwhen evaluated in the x , xy, and yx directions. The (δT 2)|y values do not follow the r2/3

law quite as closely as the others, apparently due to the influence of wind shear. However,even (δT 2)|y scales as approximately r2/3, indicating that the turbulence can be consideredto be approximately within the inertial subrange in the separation distance range marked bythe black lines in Fig. 6 (20–160 m).

The slight anisotropy of the turbulence introduces the question of whether the calculationof C2

T is appropriate. However, it would be natural to expect that wind shear of some magni-tude will be present (possibly causing anisotropy of the turbulence) in most field experimentsdesigned to investigate C2

T . Thus, it was decided that within the range 20 m ≤ r ≤ 160 m,the evaluation of the C2

T from the simulated flow data is justified. It must be noted that sinceonly a subset of the entire LES domain was utilized for these experiments, there are obvi-ously a greater number of available temperature differences at smaller separation distances,and few temperature differences at the largest separations. The relatively fewer numbers of

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100 101 102 103 10410−6

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/3]

UAS simUAS dataLES x−diffLES y−diffLES xy−diffLES yx−diff

Fig. 7 Variation of C2T with separation distance at a height of 200 m a.g.l. The black lines mark the bounds

between which the turbulence from the LES is within the inertial subrange, and the red lines show the inertialsubrange identified from the UAS measurements

temperature pairs contributing to the (δT 2) (and thus C2T ) at the largest values of r suggests

lower confidence in these values than those for small r .The C2

T values were calculated from both the LES flow fields directly and from the outputof the UAS simulator. Corresponding C2

T values were also calculated from the actual UASand sodar data recorded during the KAEFS field experiment. The presence of the inertialsubrange in the UAS turbulence data is examined by comparing C2

T at a range of separationdistances. For separation distances to be considered within the inertial subrange, C2

T values

should be approximately constant with separation distance (assuming that (δT 2) ∝ r2/3). TheC2

T values derived from the UAS flight data (red line in Fig. 7) show considerable variationwithin the inertial subrange identified from the LES C2

T data (black vertical lines in Fig. 7),as do the C2

T values calculated from the UAS simulator. The inertial subrange that couldbe seen in the UAS C2

T measurements was also identified and is marked on Fig. 7 by thevertical red lines. Within the UAS-derived inertial subrange, the C2

T measurements vary by nomore than 12 %. The bounds of the inertial subrange identified from these two measurementsets clearly differ, with the LES data showing the lower bound of the inertial subrange at aconsiderably lower value (20 m) than the UAS data (75 m). One reason for this discrepancyis the 3-s time constant of the UAS temperature sensor. Since the UAS is travelling at anairspeed of 15 m s−1, this results in unreliable estimates of C2

T below separation distancesof 45 m. This is clearly a limitation of the particular temperature sensor used on the UAS;the effect of the temperature sensor time constant is discussed further below. Although sodarreturns result from Bragg scattering related to a particular separation distance, we do notassign a separation distance value, and so the sodar-derived C2

T data are examined as verticalprofiles only. Assigning a separation distance to the C2

T data derived from the sodar wouldresult in an unfair comparison, since the other methods used in this study evaluate C2

T overhorizontal planes only, while the sodar-derived C2

T also contains contributions from verticaltemperature differences. It is clearly seen in Fig. 7 that the values of C2

T calculated fromthe recorded UAS data are approximately an order of magnitude smaller than the valuescalculated from the simulated UAS and directly from the LES data. Two potential reasonsfor this disparity are outlined below.

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4 4.5 5 5.5 6 6.5 7 7.5 813.5

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β = 0 sβ = 3 sβ = 10 sUAS obsUAS obs, β removed

Fig. 8 Upper panel the simulated temperature measurements at 200-m height with time constants of β = 0(black), β = 3 s (blue), and β = 10 s (grey). The vertical dashed lines mark the time the simulated UAS was at200-m height. Lower panel the resulting C2

T values from the temperature data in the upper panel. The red line

shows C2T from the observed UAS data, and the green line shows C2

T calculated from the UAS data when the

time constant has been corrected for. The slope of the black dashed line shows the relationship between C2T

and separation distance that results from a horizontally isothermal atmosphere

The first possible reason is associated with the time constant of the UAS temperaturesensor. The time constant of the temperature probe represents the time it takes the sensor tofully respond to a new temperature value. The UAS temperature sensor was experimentallycalibrated, finding a time constant of 3 s for an airspeed of 15 m s−1 (corresponding to theairspeed used by the UAS during the flights on April 24). The C2

T values calculated using theUAS simulator do not include the effects of the time constant on the simulated temperaturemeasurements. To examine this effect, a second experiment was performed using the UASsimulator, in which a time constant was included. The effect of the time constant was includedin the simulated temperature measurements as

Tβ(t + 1) = Tβ(t) + (1 − exp (−�t/β)) [T (t + 1) − Tβ(t)], (16)

where Tβ(t) represents the temperature with the inclusion of the time constant at time t , �tis the timestep used in the UAS simulator, β is the time constant, and T (t) is the originaltemperature measurement at time t (without the time constant included). Two values forthe time constant were tested, β = 3 s, and an increased value of β = 10 s. These β valueswere chosen since experimental calibration of the UAS temperature sensor found that a timeconstant of 3 s is applicable for an airspeed of 15 m s−1, which was the airspeed of the UASduring the measurements, and the 10-s β value is used to illustrate the effect of the timeconstant at low airspeeds. The airspeed of the simulated UAS was 18 m s−1, for which a timeconstant of 2.8 s is applicable (determined through experimental calibration of the sensor).

The effect of the inclusion of the time constant is illustrated in Fig. 8. The upper panelshows the temperature measurements obtained at a height of 200 m for the cases with no

123


time constant included (black line) and with time constants of 3 s (grey line) and 10 s (blueline). It is clear that the addition of the time constant acts to smooth the temperature data, andthis smoothing is most noticeable when the UAS moves from one height to the next (markedin the upper panel of Fig. 8 by the dashed lines). It follows that reducing the temperaturevariability in this manner will result in lower values of C2

T . This reduction is demonstratedin the lower panel of Fig. 8, which shows the C2

T values resulting from the temperature datain the upper panel. The C2

T values decrease considerably when a time constant is introduced,and the magnitude of this reduction increases for a larger time constant. Such an effect may beexpected, since the addition of the time constant acts to reduce variability in the temperaturedata. As can be seen in the lower panel of Fig. 8, the addition of the time constant has thegreatest effect on C2

T values at the smallest separation distances with less effect at largerseparation distances. It should be noted that during the calculation of C2

T from temperaturedata including a time constant, the first β+ 1 points were excluded from the temperature timeseries at each height in order to remove contamination of the data by temperatures from theprevious height. The inclusion of β in the temperature data also affects the constancy of C2

Twith separation distance, due to the smoothing of the temperature data. With an increasedvalue of β, the temperature variability tends to zero. For such a constant temperature field,(δT 2) is zero across all separation distances. Thus, the increase of the time constant translatesto a r2/3 variation of C2

T with separation distance. The dashed line in the lower panel of Fig. 8illustrates this relationship, and it is seen that the addition of the time constant brings the C2

Tvariation with separation distance closer to the relation for isothermal conditions.

The measured temperature time series from the UAS was also revised to attempt to mitigatethe effect of the 3-s time constant. To do this, we applied a simple first-order correction usingEq. 16 rearranged as

T (t + 1) = Tβ(t + 1) − Tβ(t))

(1 − exp (−�t/β))+ Tβ(t). (17)

The effect of applying this correction factor on the resulting C2T is shown by the green

line in the lower panel of Fig. 8, where the original C2T values from the UAS data (without

the correction factor applied) are shown in red. All of the data displayed in the lower panelof Fig. 8 account for advection correction. It is seen that removing the effect of the timeconstant (albeit using a rather crude method) does act to increase C2

T retrieved from the UASmeasurements, most notably at smaller separation distances. We note that this correctionfactor should only be applied if the time constant of the temperature sensor is known andcalibrated for the airspeed of the UAS.

While it is seen in Fig. 8 that the inclusion of the time constant decreases C2T , bringing

the simulated C2T values closer in line with the UAS observations, the observed C2

T values donot demonstrate the same separation-distance dependence as the simulated C2

T values withnon-zero β. When the effect of the time constant is removed from the UAS temperature data,the C2

T values are increased, bringing them closer in line with those from the UAS simulator.The separation distance dependence of C2

T also changes when the UAS data are correctedfor the time constant, coming closer to the flat line expected from the r2/3 law in the lowerpanel of Fig. 8. This suggests that a broader portion of the true inertial subrange could becaptured if the UAS data are corrected to account for the time constant. However, furtherstudy would be needed, as the correction applied here is a simple first-order equation thatdoes not replace the full temperature variability removed by the effect of the time constant.

A second possible reason for the reduced observed C2T values compared to the values

calculated from the LES is that local effects (i.e., site-specific characteristics such as small-

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Fig. 9 Dependence of C2T on

separation distance for the LESdata at a height of 200 m. Thegrey lines represent 2.4-mintemporal averages taken from 24min of LES output, while theblack line represents an averageof the full 24 min of data used inthe UAS simulator

0 100 200 30010−5

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CT

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2m

−2/3

]

scale topographical effects, flow blockage by vegetation, spatially-varying roughness lengthscaused by differing land surfaces and vegetation) are not adequately captured by the relativelycoarse nudging data provided by the WRF model. This may result in the LES and UAS datanot representing exactly the same meteorological situation.

Another limitation of using the UAS to measure C2T is associated with the time period over

which measured C2T values are representative of the atmospheric conditions. The availability

of the LES data enables an examination of the variation of C2T over different time periods

during the simulation. Figure 9 illustrates this variation, showing values of C2T calculated

from LES data over the course of the 24-min period used to populate the UAS simulator. Thegrey lines represent spatial averages across the domain, at a height of 200 m, with a 2.4-minaveraging time, which is the time period during which the simulated UAS remains at a singleheight. The black line represents C2

T values obtained over the 24-min period, which is equalto the duration of the simulated UAS flight.

It is clear from Fig. 9 that the 24-min C2T values provide a good measure of the thermal

variability over the range of scales examined by the simulated UAS, i.e., for r = 25–250 m.However, the 24-min C2

T values cannot capture the variation of the temperature within therepresentative range of its boundary-layer variability. It is also seen in Fig. 9 that the 2.4-minvalues of C2

T vary within an order of magnitude during the 24-min period. The high spatialand temporal variations of C2

T , especially in the convective boundary layer, are a well-knownfeature that has been examined in detail for C2

T derived from LES of the CBL by Cheinetand Siebesma (2009).

The wide range of measured C2T values over that 24-min period also suggests that slight

discrepancies between the C2T values from the UAS simulator and directly from the LES

data should be expected. The UAS simulator provides a C2T measurement at each height that

represents an average taken over 2.4 min, while the values derived from the LES directlyrepresent a 24-min time interval.

Aside from examining the variation of C2T with separation distance, use of the flight plan

in Fig. 1 allows for an examination of the variation of C2T with height. The vertical profiles of

C2T at 50-m separation distance, both from the LES data sampled in different directions and

the UAS simulator, are, again, similar in magnitude (Fig. 10, left panel). Minor differencesamong the profiles are primarily due to the different averaging times used by each method,and due to the smaller areal coverage of the UAS simulator. The simulated UAS spends 2.4min at each height of interest, while in the direct C2

T evaluation from LES an averaging time

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10−5 10−4 10−3 10−2 10−1 1000

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350H

eigh

t [m

]

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Time [HHMM LT]

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ght [

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LESUAS sim

Fig. 10 Left panel the variation of C2T with height via all the examined calculation methods. The separation

distance used in the both the LES and UAS-based calculations is 50 m. Centre panel sodar power return forthe duration of the UAS flight. Right panel profiles of temperature variance across the whole LES domain(grey) and across the subset samples by the UAS simulator (black)

of 24 min for each height is used. For C2T derived from UAS data, the entire profile shows

smaller C2T values compared to the values provided by the other methods. Even when the same

averaging time is used at each height (i.e., when the LES data are averaged across the domainonly for the duration of the simulated UAS flight at that height), there are still differences inthe temperature variability due to sampling a greater number of spatial points in the directLES C2

T evaluation than in the UAS simulator. This feature is illustrated in the right panelof Fig. 10, which shows the temperature variance calculated from the UAS simulator data(black line) and the LES (grey line) over the 2.4-min periods of the simulated UAS remainingat each height. It can be seen that only sampling a subset of the LES domain at each height(i.e., the circular flight path rather than the whole plane) creates a reduction in the measuredtemperature variance. Differences are also seen between the simulated measurements from theLES and UAS simulator and those from the UAS data. As discussed previously, two possiblecauses for this are the time inertia of the temperature sensor, and possible disparity betweenmeteorological conditions reproduced by the LES and captured by the UAS observations.The profile generated by correcting the measured UAS data for the time constant is shown bythe red dashed line, and the time constant removal is again seen to increase the value of themeasured C2

T , although it does not alter the shape of the profile (since the same time constantremoval is applied at each height).

The sodar model used for this experiment provides returned power in units of dB, withan arbitrarily set zero value (illustrated for the case in this study in the right panel of Fig.10). Typical values of returned power are between 40–80 dB. Directly using these valuesresults in unphysically high C2

T values, so a multiplicative constant is applied to bring C2T

down to a reasonable level. The same constant is used for all times and heights, so that C2T

profiles remain self-consistent. This implies that comparisons in vertical variations of C2T

from the different instruments are meaningful, but evaluation of C2T using the sodar method

should be performed with this consideration in mind. The noted correction does not affectany of the other methods used in this study, which employ directly measured (or modelled)temperature values to calculate C2

T . For this reason, we suggest that C2T values derived from

uncalibrated sodars should be used either for monitoring temporal changes in C2T at a certain

123


height, or for comparing vertical profiles of C2T over time. The C2

T profiles derived fromUAS and sodar data (the red and cyan lines, respectively, in Fig. 10) agree very well. Greaterconfidence is placed in the C2

T values in the lower portion of the sodar-derived profiles, asthe sodar returned power decreases sharply with height, as seen in the right panel of Fig. 10.In contrast, equal confidence is placed in the UAS data across the height range examined.The good agreement across the height range examined by the UAS and sodar (50–300 m),both of which use data from the field experiment, provides a measure of confidence in eachmethod.

6 Conclusions

One of the goals of the reported study was to test how accurately a UAS platform can determinethe structure-function parameter of turbulent temperature fluctuations, C2

T . This capabilitywas investigated through the use of a UAS simulator, which incorporates numerical LESdata to reproduce temperature measurements from the UAS. The same processing techniqueswere implemented on the simulated UAS data to calculate C2

T as are used on the actual UAStemperature measurements.

A major caveat in the method utilized is that when comparing C2T profiles and separation-

distance dependence between the measured and simulated UAS, a direct comparison cannotbe easily carried out. Discrepancies between the atmospheric state during the field experimentand the state reproduced by the LES appear to cause differences between the C2

T valuesderived from the real and simulated UAS measurements. It is also shown that the additionof an appropriate time constant to the UAS simulator reduced the discrepancy between themeasured and simulated UAS C2

T values. The time constant acts to reduce C2T values, due to

its temporal smoothing effect upon measured temperature, and should be accounted for whenusing UAS observations for examining C2

T . A simple method for removing the effects of thetime constant in the UAS data is described, but this method relies on accurate knowledgeof the time constant for the airspeed at which the UAS flight is conducted, and it maynot accurately account for the full temperature variability that is lost through inertia of thetemperature sensor.

Comparisons between C2T from the UAS simulator and LES show good agreement in both

height and separation distance (Figs. 7, 10). This indicates that the method used to calculateC2

T from UAS data in this study can provide results that are realistic and representative ofthe atmospheric state reproduced by LES. Once the technique was validated using the UASsimulator, it was tested on temperature measurements from a UAS during a field experimentin central Oklahoma. Height profiles of C2

T values derived from the UAS data compare wellin shape to those derived from a co-located sodar. Further details of the field experiment, andan in-depth observational analysis of the UAS C2

T data are presented in Bonin et al. (2015).We suggest that while UAS temperature and location data can generally be used to esti-

mate C2T , these data should be employed keeping the following caveats in mind. It must be

recognized that the time spent at each height by the UAS is relatively short, particularly forthe purpose of the statistically representative characterization of turbulence under convectiveconditions, and that derived C2

T values are representative only for the time period at eachheight. Given the wide variation of C2

T within a 24-min period shown in Fig. 9, it shouldbe kept in mind that C2

T values derived from UAS measurements are probably not generallyrepresentative of characteristic temperature-variation scales at the measurement height. Thetime constant of the temperature sensor should also be accounted for, as its inclusion canhave a large impact on the derived C2

T values, as is demonstrated in Fig. 8. Accounting for the

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advection of turbulence past the UAS is also important, particularly for cases during whichwind speeds are high. For this study, such advection was taken into account in the calcula-tion of the separation distance. For a more complete discussion on the effect of advectionof turbulence upon C2

T values computed using UAS data, see Bonin et al. (2015). With theaforementioned considerations kept in mind, UAS offers a promising platform for providingin-situ measurements of C2

T in the boundary layer over short time periods.

Acknowledgments The National Science Foundation (NSF) is acknowledged for the support of the reportedstudy through the Grant ATM-1016153. The authors acknowledge the three anonymous reviewers whosesuggestions substantially improved the manuscript.

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